Partially Hyperbolic Dynamics with Quasi-isometric Center
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913580171919360 |
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| author | Feng, Ziqiang |
| author_facet | Feng, Ziqiang |
| contents | We consider the class of partially hyperbolic diffeomorphisms on a closed 3-manifold with quasi-isometric center. Under the non-wandering condition, we prove that the diffeomorphisms are accessible if there is no $su$-torus. As a consequence, volume-preserving diffeomorphisms in this context are ergodic in the absence of $su$-tori, thereby confirming the Hertz-Hertz-Ures Ergodicity Conjecture for this class.
We show the existence of transitive Anosov flows on a closed 3-manifold admitting a non-wandering partially hyperbolic diffeomorphism with quasi-isometric center and fundamental group of exponential growth. Furthermore, we provide a complete classification of these diffeomorphisms, showing they fall into two categories: skew products and discretized Anosov flows. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_11836 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partially Hyperbolic Dynamics with Quasi-isometric Center Feng, Ziqiang Dynamical Systems Geometric Topology We consider the class of partially hyperbolic diffeomorphisms on a closed 3-manifold with quasi-isometric center. Under the non-wandering condition, we prove that the diffeomorphisms are accessible if there is no $su$-torus. As a consequence, volume-preserving diffeomorphisms in this context are ergodic in the absence of $su$-tori, thereby confirming the Hertz-Hertz-Ures Ergodicity Conjecture for this class. We show the existence of transitive Anosov flows on a closed 3-manifold admitting a non-wandering partially hyperbolic diffeomorphism with quasi-isometric center and fundamental group of exponential growth. Furthermore, we provide a complete classification of these diffeomorphisms, showing they fall into two categories: skew products and discretized Anosov flows. |
| title | Partially Hyperbolic Dynamics with Quasi-isometric Center |
| topic | Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2411.11836 |